Here is the full list of all 26 Legends of the Universe: StarCore achievements.
Activate all Teleport Pods
Acquired the Rocket Upgrade
Acquired the Long Shot Upgrade
Acquired the Slide Upgrade
Acquired the Run Upgrade
Acquired the Double Jump Upgrade
Acquired the Bomb Upgrade
Acquired the Dash Upgrade
Acquired the Strength Upgrade
Acquired the Charge Upgrade
Acquired the Water Walk Upgrade
Acquired the Heavy Rockets Upgrade
Acquired the Super Strength Upgrade
Acquired the Silver Armor Upgrade
Acquired the Gold Armor Upgrade
Find the StarCore
Defeated the Tank
Defeated Dran
Defeated Kuran
Defeated Drecht
Destroyed the Shields System
Destroyed the Engines System
Destroyed the Weapons System
Destroyed the Radar System
Destroyed the Navigation System
Get 999 Zil